Limits (essay)

essay prompt: Is the idea of 'limit' indispensable? Neutral? A hindrance?

I contend that some concept of 'limit' is indispensable in mathematics, physics, and beyond. I will make this case inductively, beginning with an exposition of the facts of reality that gave rise to the need for such a concept, and concluding with a proper definition of 'limit'.

[TODO maybe also talk about invalid versions of the concept?]

Fact of reality 1:

The Earth is a sphere, but its radius is so large, and the circumstances of everyday life take place on a scale so small, that in most contexts, it is valid to think of the Earth as a plane.

We may say, informally, that a plane is the limit of a sphere as its radius "tends towards infinity". [TODO one has to be careful about which limit one is taking, TODO harry doesn't like the word infinity, but I don't know how else to say it]

We may also say that in the context set by many everyday situations, the difference between a plane, and a sphere with 4000mi radius, is nill.

One might ask: why would I ever treat the Earth as a plane when it is in fact a sphere? What's the point of passing to the limit? The point is that it helps the crow. Here are some hypothetical examples:

• You might find it useful to know that the angles of a triangle add up to 180˚. However, this fact is only valid for triangles on a plane; "triangles" on a sphere can have angles that add up to more than 180˚. If you are thinking of your back yard as a sphere, then you have to go through an extra step, where you first think: the size of this triangle is very small relative to the sphere, and therefore its angles add up to something very close to 180˚. This is true, but it is extra cognitive work, which can be avoided by thinking of your back yard as a plane.
• You might find it useful to know that the two parallel lines you drew in the ground will not intersect. However, this fact is only valid for parallel lines on a plane; "parallel lines" on a sphere will eventually intersect. If you are thinking of the ground as a sphere, then you will have to go through an extra step, where you first think: the point at which the curvature of the earth will cause these line to intersect lies many miles out into the Pacific Ocean, and so it's completely irrelevant for my goal of making a pretty design (or whatever the goal is). Again, that is true, but it is extra cognitive work, which can be avoided by thinking of the ground as a plane.

Fact of reality 2:

[TODO write neatly] The sum from j = 0 to N of 2^{-j} is tabulated below for various values of N

  N   sum

----|-----

  0   1

  1   1.5

  2   1.75

  3   1.875

  4   1.9375

  5   1.96875

  6   1.984375

    ...

  30  1.999999999068677425384521484375

It appears that the larger N gets, the closer this sum gets to 2.

One might ask: How do we really know that if we continue for long enough, we'll get increasingly closer to 2? All we've seen is that we get very close to 2 after we sum the first 30 numbers. But maybe something unexpected will happen if we sum up to 3000, 300, or even 31.

Such a question can be answered as follows. First, note that

  (1 + x + x^2 + ... + x^N)(1 - x) = 1 + x + x^2 + ... + x^N

                                       -(x + x^2 + ... + x^N) - x^{N+1}

                                   = 1 - x^{N+1}.

Therefore, rearranging, we find

  1 + x + x^2 + ... + x^N = (1 - x^{N+1})/(1 - x)

or for x = 1/2,

  1 + 1/2 + 1/2^2 + ... + 1/2^N = 2 - 2^{-N}.

Thus we have precisely quantified the difference between (the sum up to N) and 2; it is 2^{-N}. The larger our value of N gets, the smaller this difference will become.

In fact, for any nill value epsilon, there exists some N such that the difference between (the sum up to N) and 2 is nill.

--

One might ask: Who CARES about passing to the limit? What do we gain from talking about 2 instead of 1.984375 ? The answer is that the former is simpler than the latter.

I will introduce the principle of Occam's razor,[TODO]

Real example?

Why might you actually have to do this sum? Well let's say you're a contractor for a new airport, and they picked a design for the marble tiles that looks like a spiral: [TODO show picture]. Each tile has an area of 1 square meter. How much marble do you need per tile?

One way of thinking about it is that each tile is made up of a sequence of subtitles. The first subtile will be made of 1/2 of a square meter of marble, the second 1/4th of a square meter marble, the third 1/8th of a square meter of marble, and so on. You could add those quantities up for each subtile, and find that you need

$$1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256=0.99609375$$

square meters of marble, then round that up to 1 square meter of marble. Another, easier way of thinking about it, is to note that the pattern basically fills up the entire square meter (which is perceptually obvious in this case), so therefore you need 1 square meter of marble per tile.

Fact of reality 3: [TODO]

This one should be something having to do with a practical application of a derivative or an integral.

one idea is find the average height of a sine wave. This is contrived (when would I ever do it IRL?), and it's something you could figure out without calculus or limits.

starting over

I contend that some concept of limit is indispensable in mathematics, but that one must be very cautious in using standard mathematics' concept of limit.

Outline:

1. Explain what a limit is (on my view)
2. Explain why limits are useful, e.g. by giving examples of limits
3. Explain standard mathematics' definition of limit, and the problems that I think it has

What is a limit?

In this section, I will explain what a limit is. A limit can only be taken with respect to a sequence, so first, I must explain what a sequence is.

A sequence is a function from the natural numbers to a group $C$ of existents, where there is some notion of closeness or distance between the units of $C$. A sequence is like a set of instructions: you give me any natural number $n$, and I'll show you how to produce or identify some thing $a_{n}$, where $a_{n}$ is a $C$. If the sequence consists of decimal numbers, then another way of thinking of the sequence is instructions for continuing the "...". Some examples of sequences:

1. "$a_{1},a_{2},a_{3},\cdots$", where $a_{n}:=\sum _{i=1}^{n}2^{-i}$.
2. "$b_{1},b_{2},b_{3},\cdots$", where $b_{n}$ is a circle of radius $n$, centered at $(x,y)=(n,0)$.
3. "$c_{1},c_{2},c_{3},\cdots$", where $c_{n}$ is a secant line to the graph of $f(t)=10-10t^{2}$ near $t=1$. More formally, $c_{n}$ is the unique line passing through the points $\left(1+1/n,f(1+1/n)\right)$ and $\left(1-1/n,f(1-1/n)\right)$.
4. "$d_{1},d_{2},d_{3},\cdots$", where $d_{n}:=n$.
5. "$e_{1},e_{2},e_{3},\cdots$", where $e_{n}:=\sum _{k=0}^{n}{\frac {(2k+1)!}{2^{3k+1}(k!)^{2}}}$.

Sometimes, but not always, sequences have the property that the elements of the sequence get closer and closer to a fixed point $L$ as the sequence progresses. In such cases, the sequence is said to converge to $L$, and $L$ is said to be the limit of the sequence. Examples:

1. The numbers "$a_{1},a_{2},a_{3},\cdots$" get closer and closer to 1, as $n$ gets larger and larger.
2. The circles "$b_{1},b_{2},b_{3},\cdots$" gets closer and closer to being the y-axis (from the perspective of someone sitting at the origin), as $n$ gets larger and larger.
3. The lines "$c_{1},c_{2},c_{3},\cdots$" get closer and closer to being the tangent line to the graph of $f(t)$ at $t=1$, as $n$ gets larger and larger.
4. The numbers "$d_{1},d_{2},d_{3},\cdots$" get larger and larger as $n$ gets larger and larger. The sequence does not converge; it does not have a limit.
5. The sequence "$e_{1},e_{2},e_{3},\cdots$" converges to ${\sqrt {2}}$. That is, as $n$ gets larger and larger, $e_{n}^{2}$ gets closer and closer to 2.

I will conclude with a concept-definition of a limit: The limit of a sequence is the place it goes to once we are well beyond the level of precision that is relevant in our given context.

Although the reader should keep in mind that the concept of limit extends far beyond the realm of numbers, for simplicity I will henceforth only consider sequences of numbers.

Why are limits useful?

In this section, I will give some fictional but realistic scenarios in which limits could be used to great effect. [TODO]

1. Converging to 1

example with the design in a square

2. circles

earth is like a plane locally

3. tangent lines

A typewriter[?TODO] is dropped from a height of 10 meters, and we wish to know if it will break when it hits the ground. As an intermediate step, we must determine how fast it will be moving when it hits the ground.

The height of the typewriter[TODO], as a function of time, is $h(t)=10-10t^{2}$, and it hits the ground after $t=1$ second has passed.

To find the speed of the typewriter as it hits the ground, we could find the slope of the line $c_{n}$ for some large $n$. For example, the slope of $c_{12}$ is

5. square root of 2

blahblahblah

criticism

Well, why do we have to pass all the way to the limit? Can't we just pick a large n and say that that's good enough?

The reason why is that limits simplify things. If you're designing a fence for your back yard, it is inc

What are the problems with limits?

[TODO] in a sense, the problem comes in when you have to make a formal definition. I don't know if it's actually a problem with the formal definition, or if it's just that formal definitions encourage you to drop context.